A registered result in its source context

WOIT.BETA3.DUALITY.01

Original ledger classification: Identity [E] (exact sympy algebra on generic block parameters -- block theorem, branch dichotomy, signature-mirror theorem, kill emptiness, unique induced member, exact rational-pi placement arithmetic, exact lattice regressions) + two 40-digit inertia certificates + [C] (the declared conventions: seam line, seam preservation, RP-side placement, torsor relabeling; contract-level identification, not a derivation); verdict SUCCESS, kill branch empty by algebra, no marker moves

Original claim text

THE WOIT-BETA3 MILESTONE EXECUTED (2026-07-30; one module, 22 checks, ~2 s; probe woit_os_beta3_pt_dual_probe.py 25/25, verdict SUCCESS per the frozen preregistration): 'the PT <-> PT* duality typed against sigma_std' -- contract wording: success = Woit's Minkowski conjugation identified with the family-D conjugation on the quotient; kill = the duality forces a clock-centralising (family A) structure. (1) THE COMPATIBLE CLASS (block theorem): a duality is a nondegenerate Hermitian form Phi on C^4 (sigma(Z) = Z^dagger Phi); clock compatibility (rho4 = diag(i,1,i,1), the v519 W-S4 blockwise lift) has entrywise residual (conj(lam_j) lam_k - 1) Phi_jk and forces the four cross-eigenspace entries to ZERO: two 2x2 Hermitian blocks on {e1,e3} (+) {e2,e4}, 16 -> 8 real parameters; deck compatibility adds nothing; the wrong clock lift diag(i,i,1,1) forces a different split (the compiler clock is load-bearing). (2) THE DICHOTOMY + THE FORCED (2,2): the rho_tw pairing M_tw^dagger Phi M_tw = eps conj(Phi) (rho_tw = Woit's Euclidean structure, v519 W-S4.1) splits the class exactly in two -- EUCLIDEAN (eps = +1; sigma_std = Phi = 1 lives here): seam-line norm a(|z|^2 + 1), NO null point, no OS cut ever; MINKOWSKI (eps = -1): seam-line norm a(|z|^2 - 1) -- the null locus meets the seam line EXACTLY in the seam circle |z| = 1 for EVERY member (the OS cut is FORCED, not imposed) -- and Phi_1 = -conj(Phi_i) exactly (charpoly_1(x) = charpoly_i(-x)): the TOTAL SIGNATURE IS (2,2) for every nondegenerate member -- Woit's Minkowski signature is a THEOREM of clock + rho_tw pairing, not an input. (3) THE KILL IS EMPTY BY ALGEBRA: s_Phi = Phi^T o conj inverts clock AND deck identically in the block parameters (family D, both branches); family A would need rho4^-1 = lambda rho4, i.e. rho4^2 = scalar, but rho4^2 = diag(-1,1,-1,1): NO solution -- decided, not scanned; s_Phi^2 = diag(a^2,a^2,c^2,c^2) > 0: the Kramers-free class. (4) THE INDUCED MEMBER IS UNIQUE AND PERPENDICULAR TO sigma_std: seam preservation <=> b = 0; every seam-preserving Minkowski member induces the SAME projective conjugation z -> -conj(z) (mu = -1, fixed points +-i), perpendicular to sigma_std's z -> +conj(z) (mu = +1, fixed points +-1); the clock conjugation exchanges exactly these two (mu -> -mu) -- the projective shadow of theta_cut o theta_perp = alpha_{N/2}. (5) THE QUOTIENT IDENTIFICATION: exact placement arithmetic (cut axis k = 7 fixes bond midpoints {7pi/8, 15pi/8}, perp axis k = 3 fixes {3pi/8, 11pi/8}; offsets exactly -+pi/2 -- under the v519 precision-(ii) RP-side placement the perp axis carries the +-i mark pair) + coordinate transport (axis angle 11pi/8 = 3pi/8 mod pi): THE INDUCED MINKOWSKI MEMBER IS theta_perp, the mirror whose descent v524 certified (Theta_phys) -- sigma_std's member transports to the CUT axis (the OS metric itself): role separation complete; v524 regressions re-verified at N = 8 (Gram PD (16,0,0) min 3.3471e-3 at 40 digits, theta_perp anti-unitary wrt the Gram exactly with eta_perp = +i, Theta_phys^2 = +1 on BOTH parity sectors, time reversal exact, torsor closure exact at support level). (6) CONTROLS: family A has no quotient ((4,4,0) one-particle, v524 C4.2 verbatim); the Euclidean branch cannot cut; a non-mu4 reflection (e^{2 pi i/5}) is not mark-anchored. NAMED LIMITS AS LOAD-BEARING CONTENT: free/quotient level ONLY -- nothing constructs the interacting algebra A_hol, all seven contract kill tests stay formally live there, WOIT.OS.TWISTOR.01 stays [O], gamma (the chirality theorem + the mark incidence) is the next milestone; the declared [C] conventions (seam line span(e1,e2) with the v519 blockwise lift; 'the duality respects the seam' read as seam-line preservation; the RP-side bond-midpoint placement, v519 precision (ii); the clock relabeling of the torsor) are cited from v492/v519/v524, not invented; 'identified' means inside these declared compatibility classes, NOT a derivation of Minkowski space; the word 'derived' appears nowhere for any physics statement.

Original classification detail

[E] block theorem (4 cross entries forced zero; 16 -> 8 params); dichotomy eps = +-1 with the OS cut forced on the Minkowski branch (seam norm a(|z|^2 - 1)) and signature (2,2) by the mirror theorem; kill empty (rho4^2 = diag(-1,1,-1,1) not scalar); induced member unique z -> -conj(z) = theta_perp = the v524 Theta_phys carrier (placement offsets exactly -+pi/2; axis 11pi/8 = 3pi/8 mod pi); N = 8 regressions: Gram PD (16,0,0) min 3.3471e-3, eta_perp = +i, Theta^2 = +1 both sectors, time reversal + torsor closure exact; family A (4,4,0); 22/22 checks

Dependencies as recorded

WOIT.OS.TWISTOR.01; WOIT.THETA.FREE.01; WOIT.BETA1.GSO.01; WOIT.BETA2.OS.01; SEAM.BIT.RPBLIND.01; v492; v519; v522; v524